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On the algebraic structure in Markovian processes of death and epidemic types

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Philippe Picard  and
Claude Lefèvre
Philippe Picard*
Affiliation:
Université de Lyon 1
Claude Lefèvre*
Affiliation:
Université Libre de Bruxelles
*
∗ Postal address: Mathématiques Appliquées, Université de Lyon 1, 43 boulevard du 11 Novembre 1918, F-69622 Villeurbanne Cedex, France.
∗∗ Postal address: Institut de Statistique et de Recherche Opérationnelle, C.P. 210, Université Libre de Bruxelles, Boulevard du Triomphe, B-1050 Bruxelles, Belgique. Email address: clefevre@ulb.ac.be
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Abstract

This paper is concerned with the standard bivariate death process as well as with some Markovian modifications and extensions of the process that are of interest especially in epidemic modeling. A new and powerful approach is developed that allows us to obtain the exact distribution of the population state at any point in time, and to highlight the actual nature of the solution. Firstly, using a martingale technique, a central system of relations with two indices for the temporal state distribution will be derived. A remarkable property is that for all the models under consideration, these relations exhibit a similar algebraic structure. Then, this structure will be exploited by having recourse to a theory of Abel-Gontcharoff pseudopolynomials with two indices. This theory generalizes the univariate case examined in a preceding paper and is briefly introduced in the Appendix.

Keywords

Exact temporal distributionbivariate death processMarkovian SIR epidemic modelsmartingalesAbel-Gontcharoff pseudopolynomials with two indicesAbelian expansions

MSC classification

Primary: 60J75: Jump processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 31 , Issue 3 , September 1999 , pp. 742 - 757
Copyright
Copyright © Applied Probability Trust 1999 

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